<h2>DESCRIPTION</h2> 

<em>t.vect.algebra</em> performs temporal and spatial overlay and
buffer functions on space time vector datasets (STVDS) by using the
temporal vector algebra. New STVDS can be created, which are
expressions of existing STVDS.

<h3>PROGRAM USE</h3>
The module expects an <b>expression</b> as input parameter in the following form:
<p>

<div class="code"><pre>
"result = expression"
</pre></div>

The statement structure is similar to r.mapcalc, see <a href="r.mapcalc.html">r.mapcalc</a>.
Where <b>result</b> represents the name of a space time dataset (STVDS) that will
contain the result of the calculation that is given as <b>expression</b>
on the right side of the equality sign.

These expression can be any valid or nested combination of temporal 
operations and functions that are provided by the temporal vector 
algebra.<br>
The algebra provides methods for map selection from STDS based on their 
temporal relations. It is also possible to temporally shift maps, to 
create temporal buffer and to snap time instances to create a valid 
temporal topology. Furthermore expressions can be nested and evaluated 
in conditional statements (if, else statements). Within if-statements 
the algebra provides temporal variables like start time, end time, day 
of year, time differences or number of maps per time interval to build 
up conditions. These operations can be assigned to space time datasets 
or to the results of operations between space time datasets.

<p> As default, topological relationships between space time datasets 
will be evaluated only temporal. Use the <b>s</b> flag to activate the 
additionally spatial topology evaluation. <p> The expression option 
must be passed as <b>quoted</b> expression, for example: <br>

<div class="code"><pre>
t.select expression="C = A : B"
</pre></div>

Where <b>C</b> is the new space time raster dataset that will contain 
maps from <b>A</b> that are selected by equal temporal relationships to 
the existing dataset <b>B</b> in this case.

<h2>TEMPORAL VECTOR ALGEBRA</h2>

The temporal algebra provides a wide range of temporal operators and
functions that will be presented in the following section. <p>

<h3>TEMPORAL RELATIONS</h3>

Several temporal topology relations between registered maps of space
time datasets are supported: <br>
<div class="code"><pre>
equals            A ------
                  B ------

during            A  ----
                  B ------

contains          A ------
                  B  ----

starts            A ----
                  B ------

started           A ------
                  B ----

finishs           A   ----
                  B ------

finished          A ------
                  B   ----

precedes          A ----
                  B     ----

follows           A     ----
                  B ----

overlapped        A   ------
                  B ------

overlaps          A ------
                  B   ------

over              booth overlaps and overlapped
</pre></div>

The relations must be read as: A is related to B, like - A equals B - A is
during B - A contains B <p>
Topological relations must be specified in {} parentheses. <br>

<h3>TEMPORAL OPERATORS</h3>

The temporal algebra defines temporal operators that can be combined 
with other operators to perform spatio-temporal operations. The 
temporal operators process the time instances and intervals of two 
temporal related maps and calculate the result temporal extent by five 
different possibilities.

<div class="code"><pre>
LEFT REFERENCE     l       Use the time stamp of the left space time dataset
INTERSECTION       i       Intersection
DISJOINT UNION     d       Disjoint union
UNION              u       Union
RIGHT REFERENCE    r       Use the time stamp of the right space time dataset
</pre></div>

<h3>TEMPORAL SELECTION</h3>

The temporal selection simply selects parts of a space time dataset without
processing raster or vector data.

The algebra provides a selection operator <b>:</b> that selects parts
of a space time dataset that are temporally equal to parts of a second one
by default. The following expression

<div class="code"><pre>
C = A : B
</pre></div>

means: Select all parts of space time dataset A that are equal to B and 
store it in space time dataset C. The parts are time stamped maps.

<p> 
In addition the inverse selection operator <b>!:</b> is defined as 
the complement of the selection operator, hence the following 
expression

<div class="code"><pre>
C = A !: B
</pre></div>

means: select all parts of space time time dataset A that are not equal to B
and store it in space time dataset (STDS) C.<p>

To select parts of a STDS by different topological relations to other 
STDS, the temporal topology selection operator can be used. The 
operator consists of the temporal selection operator, the topological 
relations, that must be separated by the logical OR operator <b>|</b> 
and the temporal extent operator. All three parts are separated by 
comma and surrounded by curly braces:

<div class="code"><pre>
{"temporal selection operator", "topological relations", "temporal operator"}
</pre></div>

<p>

Examples:
<div class="code"><pre>
C = A {:, equals} B
C = A {!:, equals} B
</pre></div>

We can now define arbitrary topological relations using the OR operator "|"
to connect them:

<div class="code"><pre>
C = A {:,equals|during|overlaps} B
</pre></div>

Select all parts of A that are equal to B, during B or overlaps B. <br>

In addition we can define the temporal extent of the result STDS by adding the
temporal operator.

<div class="code"><pre>
C = A {:, during,r} B
</pre></div>

Select all parts of A that are during B and use the temporal extents
from B for C.
<p>

The selection operator is implicitly contained in the temporal topology
selection operator, so that the following statements are exactly the same:

<div class="code"><pre>
C = A : B
C = A {:} B
C = A {:,equal} B
C = A {:,equal,l} B
</pre></div>

Same for the complementary selection:

<div class="code"><pre>
C = A !: B
C = A {!:} B
C = A {!:,equal} B
C = A {!:,equal,l} B
</pre></div>

<h3>CONDITIONAL STATEMENTS</h3>

Selection operations can be evaluated within conditional statements.
<br>
Note A and B can either be space time datasets or expressions. The temporal 
relationship between the conditions and the conclusions can be defined at the 
beginning of the if statement. The relationship between then and else conclusion 
must be always equal.

<div class="code"><pre>
if statement                           decision option                        temporal relations
  if(if, then, else)
  if(conditions, A)                    A if conditions are True;              temporal topological relation between if and then is equal.
  if(conditions, A, B)                 A if conditions are True, B otherwise; temporal topological relation between if, then and else is equal.
  if(topologies, conditions, A)        A if conditions are True;              temporal topological relation between if and then is explicit specified by topologies.
  if(topologies, conditions, A, B)     A if conditions are True, B otherwise; temporal topological relation between if, then and else is explicit specified by topologies.
</pre></div>

<h4>Logical operators</h4>

<div class="code"><pre>
Symbol  description

  ==    equal
  !=    not equal
  >     greater than
  >=    greater than or equal
  <     less than
  <=    less than or equal
  &&    and
  ||    or
</pre></div>

<h4>Temporal functions</h4>

The following temporal function are evaluated only for the STDS that 
must be given in parenthesis.

<div class="code"><pre>
td(A)                    Returns a list of time intervals of STDS A

start_time(A)            Start time as HH::MM:SS
start_date(A)            Start date as yyyy-mm-DD
start_datetime(A)        Start datetime as yyyy-mm-DD HH:MM:SS
end_time(A)              End time as HH:MM:SS
end_date(A)              End date as yyyy-mm-DD
end_datetime(A)          End datetime as  yyyy-mm-DD HH:MM

start_doy(A)             Day of year (doy) from the start time [1 - 366]
start_dow(A)             Day of week (dow) from the start time [1 - 7], the start of the week is Monday == 1
start_year(A)            The year of the start time [0 - 9999]
start_month(A)           The month of the start time [1 - 12]
start_week(A)            Week of year of the start time [1 - 54]
start_day(A)             Day of month from the start time [1 - 31]
start_hour(A)            The hour of the start time [0 - 23]
start_minute(A)          The minute of the start time [0 - 59]
start_second(A)          The second of the start time [0 - 59]
end_doy(A)               Day of year (doy) from the end time [1 - 366]
end_dow(A)               Day of week (dow) from the end time [1 - 7], the start of the week is Monday == 1
end_year(A)              The year of the end time [0 - 9999]
end_month(A)             The month of the end time [1 - 12]
end_week(A)              Week of year of the end time [1 - 54]
end_day(A)               Day of month from the start time [1 - 31]
end_hour(A)              The hour of the end time [0 - 23]
end_minute(A)            The minute of the end time [0 - 59]
end_second(A)            The second of the end time [0 - 59]
</pre></div>

<h4>Comparison operator</h4>

The conditions are comparison expressions that are used to evaluate
space time datasets. Specific values of temporal variables are
compared by logical operators and evaluated for each map of the STDS and 
the related maps.
For complex relations the comparison operator can be used to combine conditions: 
<br>

The structure is similar to the select operator with the extension of an aggregation operator:

<div class="code"><pre>
{"comparison operator", "topological relations", aggregation operator, "temporal operator"}
</pre></div>

<br>
This aggregation operator (| or &) define the behaviour if a map is related the more 
than one map, e.g for the topological relations 'contains'.
Should all (&) conditions for the related maps be true or is it sufficient to 
have any (|) condition that is true. The resulting boolean value is then compared 
to the first condition by the comparison operator (|| or &&). 
As default the aggregation operator is related to the comparison operator: <br>

Comparison operator -&gt; aggregation operator:

<div class="code"><pre>
|| -> | and && -> & 
</pre></div>

Examples:

<div class="code"><pre>
Condition 1 {||, equal, r} Condition 2
Condition 1 {&&, equal|during, l} Condition 2
Condition 1 {&&, equal|contains, |, l} Condition 2
Condition 1 {&&, equal|during, l} Condition 2 && Condition 3
Condition 1 {&&, equal|during, l} Condition 2 {&&,contains, |, r} Condition 3
</pre></div>

<h4>Hash operator</h4>

Additionally the number of maps in intervals can be computed and used in 
conditional statements with the hash (#) operator. <br>

<div class="code"><pre>
A{#, contains}B
</pre></div>

This expression computes the number of maps from space
time dataset B which are during the time intervals of maps from
space time dataset A.<br>
A list of integers (scalars) corresponding to the maps of A
that contain maps from B will be returned.
<p>

<div class="code"><pre>
C = if({equal}, A {#, contains} B > 2, A {:, contains} B)
</pre></div>

This expression selects all maps from A that temporally contains at least 2 
maps from B and stores them in space time dataset C. The leading equal statement 
in the if condition specifies the temporal relation between the if and then part 
of the if expression. This is very important, so we do not need to specify a 
global time reference (a space time dataset) for temporal processing.
<p>
Furthermore the temporal algebra allows temporal buffering, shifting
and snapping with the functions buff_t(), tshift() and tsnap()
respectively.

<div class="code"><pre>
buff_t(A, size)         Buffer STDS A with granule ("1 month" or 5)
tshift(A, size)         Shift STDS A with granule ("1 month" or 5)
tsnap(A)                Snap time instances and intervals of STDS A
</pre></div>


<h4>Single map with temporal extent</h4>

The temporal algebra can also handle single maps with time stamps in 
the tmap function.

<div class="code"><pre>
tmap()
</pre></div>

For example:

<div class="code"><pre>
 C = A {:,during} tmap(event)
</pre></div>

This statement select all maps from space time data set A that are 
during the temporal extent of single map 'event'

<h3>Spatial vector operators</h3>

The module supports the following boolean vector operations:<br>

<div class="code"><pre>
 Boolean Name   Operator Meaning         Precedence   Correspondent function 
---------------------------------------------------------------------------------- 
 AND            &        Intersection          1      (v.overlay operator=and) 
 OR             |        Union                 1      (v.overlay operator=or)  
 DISJOINT OR    +        Disjoint union        1      (v.patch)              
 XOR            ^        Symmetric difference  1      (v.overlay operator=xor) 
 NOT            ~        Complement            1      (v.overlay operator=not) 
</pre></div>

And vector functions:

<div class="code"><pre>
 buff_p(A, size)    	  Buffer the points of vector map layer A with size
 buff_l(A, size)    	  Buffer the lines of vector map layer A with size
 buff_a(A, size)    	  Buffer the areas of vector map layer A with size
</pre></div>

<h3>Combinations of temporal, vector and select operators</h3>

We combine the temporal topology relations, the temporal operators and 
the spatial/select operators to create spatio-temporal vector 
operators:

<pre class="code">
{"spatial or select operator" , "list of temporal relations", "temporal operator" } 
</pre><p>

For multiple topological relations or several related maps the spatio-temporal 
operators feature implicit aggregation.

The algebra evaluates the stated STDS by their temporal topologies and apply 
the given spatio temporal operators in a aggregated form.

If we have two STDS A and B, B has three maps: b1, b2, b3 that are all during 
the temporal extent of the single map a1 of A, then the following overlay 
calculations would implicitly aggregate all maps of B into one result map for 
a1 of A:

<div class="code"><pre>
C = A {&, contains} B --> c1 = a1 & b1 & b2 & b3
</pre></div>

Keep attention that the aggregation behaviour is not symmetric:

<div class="code"><pre>
C = B {&, during} A --> c1 = b1 & a1
                        c2 = b2 & a1
                        c3 = b3 & a1
</pre></div>

<h3>Examples: </h3>

Spatio-temporal intersect all maps from space time dataset A with all 
maps from space time dataset B which have equal time stamps and are 
temporary before Jan. 1. 2005 and store them in space time dataset D.

<div class="code"><pre>
D = if(start_date(A) < "2005-01-01", A & B)
</pre></div>

Buffer all vector points from space time vector dataset A and B with a 
distance of one and intersect the results with overlapping, containing, 
during and equal temporal relations to store the result in space time 
vector dataset D with intersected time stamps.

<div class="code"><pre>
D = buff_p(A, 1) {&,overlaps|overlapped|equal|during|contains,i} buff_p(B, 1)
</pre></div>

Select all maps from space time dataset B which are during the temporal 
buffered space time dataset A with a map interval of three days, else
select maps from C and store them in space time dataset D.

<div class="code"><pre>
D = if(contains, td(buff_t(A, "1 days")) == 3, B, C)
</pre></div>

<h2>REFERENCES</h2>

<a href="http://www.dabeaz.com/ply/">PLY(Python-Lex-Yacc)</a>

<h2>SEE ALSO</h2>

<em>
<a href="t.select.html">t.select</a>
</em>


<h2>AUTHORS</h2>

Thomas Leppelt, Soeren Gebbert, Th&uuml;nen Institute of Climate-Smart Agriculture

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